There are very simple examples of compact dynamical systems with ergodic invariant measures whose generic points are Borel sets of low complexity:
If \(X\)  is the unit circle and \(T\)  is an irrational rotation, then every point is generic for the Lebesgue measure \(\lambda \)  so \(G_\lambda \)  is clopen. On the other hand, if \(X=[0,1]\)  and \(T(x)=x^2\)  for \(0\le x \le 1\) , then \(T\)  has two ergodic invariant measures, namely Dirac-point measures \(\delta _0\)  and \(\delta _1\)  concentrated at fixed points 0 and 1. We clearly have
\(G_{\delta _0}=[0,1)\)  is open but not closed, and \(G_{\delta _1}=\lbrace 1\rbrace \)  is closed but not open. Hence, similarly as in [1]} some assumptions about the dynamical system are necessary to achieve the maximal complexity of sets with prescribed Birkhoff averages for some (or all) observables.
It turns out that the right assumption to
impose on the system is a form of the specification property.
The specification property was introduced by R. Bowen in his
paper on Axiom A diffeomorphisms [2]}. However, for the main results of our paper
only a certain weak form of the specification property (coined the strong approximate product structure) is needed.
It implies the feeble specification property used in [1]} for symbolic systems.
We refer the reader to [4]} for a discussion of the specification property
and its many variants as well as their significance in dynamics.
